
Up to \((\frac{r_c}{r^*})^0\) : In this case, \(r_c\)  is not appear and it is expected that we reach the results corresponding to AdS black hole in the high temperature limit [1]}, [2]}. We consider the first terms in brackets in eqs. (REF ) and (REF ) which include \(L_1\)  and \(D_1\) , respectively and sum over \(m\) . We obtain

l=2r*n=014n+1(n+12)(2n+23)(n+1)(4n+16)(rHr*)4n,


A=2r*2L2n=014n-2(n+12)(2n+23)(n+1)(4n+16)(rHr*)4n.

Using eqs. (REF ) and (REF ) and doing some calculations, we get

A=2r*2L2lr*2-3(23)2(16)+n=13(4n-2)(4n+1)(n+12)(2n+23)(n+1)(4n+16)(rHr*l)4n.

It is easy to see that in the large \(n\)  limit the above infinite series behaves as \(\frac{1}{n^2}(\frac{r_H}{r^*})^{4n}\) , so we can safely consider \(r^*\rightarrow r_H\)  limit. Hence, the final result in the high temperature limit becomes

SHT(0)(lT)4GN(5)Sfinite(l,lT)L2T2= [F1+3 (lT)],        F1<0,

where \(\mathcal {S}_{HT}^{(0)}(lT)\)  is the redefined \(S_{finite}(l,lT)\)  in the high temperature limit up to the zero order of \(\frac{r_c}{r^*}\)  and \(F_1\)  is a numerical constant given by eq.(REF ) in appendix . The second term in eq.(REF ) is the dominant term in \(S_{finite}(l,lT)\)  which scales with the volume of the rectangular strip, \(lL^2\) , and the first term in \(S_{finite}(l,lT)\)  is area dependent, \(L^2\) . Therefore, the first term corresponds to the entanglement entropy between the strip region and its complement while the second term corresponds to the thermal entropy.

Up to \((\frac{r_c}{r^*})^2\) : In this case, we add the non-conformal effects to the previous one by considering the second terms in brackets in eqs. (REF ) and (REF ). we rewrite eq.(REF ) as follows

A=2r*2L2n=0m=0(n+12)(m+12)(n+1)(m+1)(rHr*)4m[L1+L2(rcr*)2+(D1-L1)+(D2-L2)(rcr*)2]

Performing the summation over \(m\) , then the above equation leads to

A=r*3L2l+2r*2L2[-3(23)2(16)+n=13(4n-2)(4n+1)(n+12)(2n+23)(n+1)(4n+16)(rHr*)4n]+2r*2L2B(rcr*)2,

where \(B\)  is given by

B=B0+n=0m=1(n+12)(m+12)(n+1)(m+1)(rHr*)4m[3 n4m+6 n-2)+3-6 n8 m+12 n
-32(2 n+1)4 m+6 n+4+32(2 n+1)4 m+6 n+6].

\(B_0\)  is a constant comes from \(m=0\)  and given by eq.(REF ) in appendix . The convergence of the series in the second term of eq.(REF ) was seen in the previous case. In order to study the convergence of the series in eq.(REF ) at high temperature, \(r^*\rightarrow r_H\) , we sum over \(n\)  and obtain
<FIGURE>B=B0+m=13(m+12)8(m+1)(rHr*l)4m[1m(  4F3(12,2 m5+45,2 m3-13,2 m3;2 m5-15,2 m3+23,2 m3+1;1))-1(m+1) (2 m+3)(  3F2(32,2 m3+23,2 m3+1;2 m3+53,2 m3+2;1))].      
It is easy to see that in the large \(m\)  limit, the first series in the brackets behaves as \(\frac{1}{m^\frac{3}{2}}(\frac{r_H}{r^*_l})^{4m}\)  and the second series in the brackets behaves as \(\frac{1}{m^\frac{5}{2}}(\frac{r_H}{r^*_l})^{4m}\) . Therefore, both of these series are convergent at \(r^*\rightarrow r_H\)  and we can safely take the mentioned limit. We finally reach the following expression

SHT(2)(lc,lT)4GN(5)Sfinite(l,lc,lT)L2c T=1(lc )(lT)[F1(lT)2+3 (lT)3+F2(lc )2],    F1<0, F2>0,     
where \(S_{HT}^{(2)}(l\Lambda _c,lT)\)  is the redefined \(S_{finite}(l,l\Lambda _c,lT)\)  and \(F_2\)  is positive and given by eq.(REF ) in appendix . The first two terms in eq.(REF ) correspond to \(S_{HT}^{(0)}\)  in the AdS black hole in the high temperature limit. The non-conformal effect appears in third term which is very small with respect to the first two terms. This term is positive and hence the non-conformal effect increases the \(S_{HT}^{(2)}(l\Lambda _c,lT)\)  with respect to \(S_{HT}^{(0)}(lT)\)  at high temperature limit. In figure REF  we plot \(S_{HT}^{(2)}(l\Lambda _c,lT)\)  as a function of \(l\Lambda _c\)  and \(lT\)  for fixed value of \(lT\)  and \(l\Lambda _c\) , respectively. From this figure, we observe that if we fix \(l\Lambda _c\)  and increase \(lT\) , then \(S_{HT}^{(2)}(l\Lambda _c,lT)\)  will increase. But in the fixed \(lT\) , \(S_{HT}^{(2)}(l\Lambda _c,lT)\)  has different behavior and decreases by increasing \(l\Lambda _c\) .

